exponen dan logaritma

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ExpoNent, Roots, & Logarithm Exponent Formula CHAPTER I

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Consider the following the problem What’s the product of 24 and 25? Solutions: am= a.a.a…..a m times 24= 2.2.2.2 =16 4 times 25=2.2.2.2.2 =32 So that the answer of the problem above is: 24.25 = 16.32 = 512 We know : 512 = 29 =16.32 =24.25 = 2 4+5 A. Exponent, Roots, And Logarithm

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If we replace the base a by a, the exponents 4 and 5 by positive integers m and n : We will get: am = a.a.a….a m times an = a.a.a….a n times am.an = (a.a….a).(a.a….a) m times n times = a.a.a……..a (m+n) times FORMULA I with condition a € R, a ? 0 a is called the base number and m,n is called exponent

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if m>n, them use formula I, we shall obtain: am-n .an = am-n+n am-n .an = am Formula II an

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pay attention the following problem : (am)n = am.am…..am n times = am+m+m….m = am.n Conclusion Formula III

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If besides a we also use the base of b, then the form (ab)m. Can be defined : (ab)m = ab.ab……ab m times = (a.a.a.a).(b.b.b.b) m times m times Formula IV

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So that by the same way we will get Conclusion : b? 0 Formula V Because : is not defined

Example : Simplify! : 

Example : Simplify!

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Answer

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B.Integers Exponents In this section we shall see that formula I-V hold for integers exponents, either positive, negative and zero If we substitute m = n use formula II we shall obtain Formula VI a € R, a ? 0

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If we subsituty m=0 use formula II we will get Formula VII a € R, a ? 0

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SIMPLIFY!! Examples

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Answer

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By the Formula I we obtain am.an = am+n If we substituting n=m, then we will get am.am = a2m so that a2m = a 2m =1 m =1/2 form a2m =a ,can be changed (am) 2 = a am = va a1/2 = va In general can written C. Fractional Exponent

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In the fractional exponents: If is changed ,then we will get : n,m is positive integers Formula VIII

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By the formula VIII: ,then if ,we obtain Formula IX Furthermore the formula I - IX,also hold for fractional exponents

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Example Simplify and write down in positive exponents!

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Simplify and write down in the roots form !

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3. Evaluate!

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4. Find the length of diagonal of rectangle if it has dimension length is cm and width is cm!

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Answer

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EXPONENT EQUATION Example : 1. Find the solution set :

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Answer

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Example : 1. Find the solution set :

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Answer

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Example : 1. Determine the solution set :

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V Answer

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: 2 V

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V V

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D. Rational Numbers and Irrational Number (1) Rational Number Rational Number are numbers which can be put in a/b form with a and b are integers numbers and b ? 0. Example: 3 coz 3 can be stated in etc 0.444…= 0.4 can be stated in 1.12 can be stated in Notes ! The lines above 12(no.3)shows that 12 is repeated unlimitedly Generally ! Rational number are repeated decimals

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(2) Irrational Numbers Irrational number are number which can’t be stated in Generally : Irrational numbers are unrepeated decimal Example! =1.4142135…. = 5.1461524…. Log 2 = 0.3010…. e =2.71828182….. coz can’t be put in

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1.Please check whether the following form are rational or irrational. Examples

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Answer

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E. Root Form Root number are the rational number which the result are irrational EXAMPLE :

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1. Please check whether the following forms are root forms or not Note : Ln = Log number with base e Ln x = elog x Excercise

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Answer

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ANSWER

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Answer

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E. Algebra Operation Of Root Form REMEMBER !!

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1. Evaluate the value of the following root forms ANSWER Examples

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Answer

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1. Find the value of the following problem !! 2. Find 2p+2q, 4pq and p2+q2 ANSWER Home Work !!!

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Answer

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G. Rationalizing The Denominator A fractional of root on its denominator such as : Can be simplified by rationalizing

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Examples

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Logarithm Formula Excercise Logarithm

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Logarithm Formula

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BACK Logarithm Formula

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ANSWER 1. EVALUATE !! Excercise

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Answer

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NEXT

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2. SIMPLIFY !! ANSWER

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Answer

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NEXT

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EVALUATE !! ANSWER

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NEXT Answer

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ANSWER

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Answer

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NEXT

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5. Find the value of x that fulfill of the following equation! ANSWER

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TM Answer

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1. Find the value of x that fulfill of the following equation !! ANSWER Home Work !!!

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Answer Ø

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V

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V

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thanks for your Attention...! cReated by: alFAth and NUgz el-Saverzs united